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Quiz 3

This document contains a statistics quiz with two questions. Question I asks to define a sample space and random variable for a coin toss experiment, find probabilities related to the number of heads, and draw the cumulative distribution function graph. Question II contains three parts asking to prove functions are probability density functions, find the corresponding cumulative distribution functions, and compute expectations and variances for random variables.

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55 views2 pages

Quiz 3

This document contains a statistics quiz with two questions. Question I asks to define a sample space and random variable for a coin toss experiment, find probabilities related to the number of heads, and draw the cumulative distribution function graph. Question II contains three parts asking to prove functions are probability density functions, find the corresponding cumulative distribution functions, and compute expectations and variances for random variables.

Uploaded by

fabas2453
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Date: 28/3/2022 339-Statistics 1442 − 1443

Deadline time: 12/4/2022 Quiz3 Maximum Marks: 5


Name: University No: Section: Serial No:

Question I: Consider an experiment in which a coin is tossed four times.


The sample space of this experiment is given as

S = {w = w1 w2 w3 w4 / w ∈ {H ( heads), T( tails)}}.

We define the random variable X for this sample space as follows:

X(w) = number of heads in sequence w.

1. How many elements are in the sample space?

2. Find, RX , the space of the random variable X

3. Find the probability density function (pdf) of X

4. Find the cumulative distribution function (cdf) of X and draw his graph

5. Use the probability distribution of X to find the following:

i) p(1 ≤ X < 6), p(X ≥ 2) and p(X ≤ 3)


ii) p(X ∈ {1, 3, 5})

Question II:

1. Consider the function


( 1
(2x + 1) if x ∈ {1, 2, 4, 6}
f (x) = 30
0 otherwise

(a) Prove that f is a probability density function (pdf) of a random variable X


(b) Find the cumulative distribution function (cdf) associated to this pdf.
(c) Compute E(X) and V (X)

1
2. Consider the function

1
6x if 0 ≤ x <


3


f (x) = 1
3(1 − x) if ≤x<1
3



 0 otherwise

(a) Prove that f is a probability density function (pdf) of a random variable X


(b) Find the cumulative distribution function (cdf) associated to this pdf.
(c) Compute E(X) and V (X)

3. Consider the function




 0 if x<0
1
12x2 if 0≤x<

F (x) = 6
− 1 + 6(x − x2 ) if 1
≤x< 1
 2 6 2


1 if x ≥ 21

(a) Prove that F is a cumulative distribution function (cdf) of a random variable


a.e. F continuous, non-decreasing, lim F (x) = 1 and lim F (x) = 0
x→+∞ x→−∞

(b) Find the probability density function (pdf) associated to this cdf.
(c) Compute p(|X| < 0.5) and p(|X − 0.5| = 1.5).
(d) Compute E(X) and V (X)

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